One can study the large-\(n\)  asymptotics of determinant \(D_n(\phi ,d \phi ,1,1)\)  (and with increasing effort \(D_n(\phi ,d \phi ,r,s)\)  for fixed \(r, s \in {\mathbb {Z}}\) ) assuming that \(\phi \)  possesses Fisher–Hartwig singularities \(\lbrace z_i\rbrace ^{m}_{i=1}\)  on the unit circle. It is in fact in this level of generality that E. Basor and T. Ehrhardt have been able to compute the asymptotics of \(D_n(\phi ,d \phi ,0,1)\) , \(D_n(-\phi ,d \phi ,0,1)\) , \(D_n(\phi ,d z \phi ,0,1)\) , and \(D_n(-z\phi ,d \phi ,0,1)\)  via the operator-theoretic methods in [1]}. However, the authors in [1]} further require that the Fisher–Hartwig part of \(\phi \)  be even. In fact they used some results of the work [3]} to prove their asymptotic formulas for Toeplitz+Hankel determinants, and for this reason they inherited the evenness assumption from the work [3]} where the authors needed evenness of \(\phi \)  in their \(2\times 2\)  setting to relate Hankel and Toeplitz+Hankel determinants to a Toeplitz determinant with symbol \(\phi \) .
